Goals from Wade Elliott, Sam Baldock and Stephen Pearson saw the Robins home and subsequently ended Boro's three-year stay in League One.
City started the brighter side and could have scored several in a dominant first half.
Stevenage keeper Chris Day twice had to deny in-form Baldock inside the first five minutes, before saving Derrick Williams' powerful eighth-minute header.
Baldock then wasted a great chance on 26 minutes, firing his shot straight at Day after being sent clean through by Elliott.
But Elliott deservedly gave the visitors the lead after 41 minutes, turning the ball in at the back post from Brendan Moloney's cross.
Baldock added a second after 65 minutes, slotting home his 24th league goal of the season.
Robins substitute Pearson then scored with his first touch on 75 minutes, coolly finishing after Tyrone Barnett's through ball, before Luke Freeman netted a late free-kick for doomed Boro.